Phase-Space Beam Summation: a Local Spectrum Analysis of Time-Dependent Radiation

Abstract

The phase-space beam summation is a general analytical framework for local analysis and modeling of radiation from extended source distributions. In this formulation the field is expressed as a superposition of beam propagators that emanate from all points in the source domain and in all directions. The theory is presented here for both time-harmonic and time-dependent fields: in the later case, the propagators are pulsed-beams (PB). The phase-space spectrum of beam propagators is matched locally to the source distribution via local spectral transforms: a local Fourier transform for time-harmonic fields and a "local Radon transform" for time-dependent fields. These transforms extract the local radiation properties of the source distributions and thus provide a priori localized field representations. Some of these basic concepts have been introduced previously for twodimensional configurations. The present paper extends the theory to three dimensions, derives the operative expressions for the transforms and discusses additional phenomena due to the three dimensionality. Special emphasis is placed on numerical implementation and on choosing a numerically converging space-time window. It is found that the twice differentiated Gaussian-window is both properly converging and provides a convenient propagator that that can readily be tracked in complicated inhomogeneous medium.

Pulsed Beam Expansion of Electromagnetic Aperture Fields

“…The lattice is overcomplete for ν x,y < 1, critically complete in the Gabor limit ν x,y ↑ 1 [45], and for ν x,y ↓ 0 the discrete parametrization attains the continuity limit as in [4,5]. It is convenient to chose equal-direction unit-cell dimensions ∆κ x = ∆κ y = ∆κ and equal-space unit-cell dimensions ∆x = ∆ȳ = ∆r t , though this choice is not essential.…”

“…Beam-type field expansion schemes have been the subject of an intense research in the past decade for scalar time-harmonic [1][2][3] as well as time-dependent fields [4][5][6][7]. The motivation to use these expansions lies in their mutual spectral-spatial (and temporal for time-dependent fields) localization and the capability to propagate the expansions' waveobjects in different complex environments.…”

“…An exact beam-type expansion of scalar planar aperture timedependent fields was introduced in [4] for 2D and in [5] for 3D configurations in which the propagating fields are decomposed into a continuous spectrum of shifted, tilted and delayed pulsedbeam waveobjects. The frame-based field expansion utilizes the overcompleteness of phase-space representations and introduced a discrete spectral representation.…”

Parameterization of the Tilted Gaussian Beam Waveobjects

“…The GB waveobject in this particular case was previously termed iso-axial [2]. The conventional (orthogonal system) GB and its time-dependent counterpart, the pulsed beam, have been well studied in [2,22] and [36].…”

“…The latter utilizes the overcompleteness nature of the beam's continuous spectrum [18,21,22], and discretizes the spectral representation with no loss of essential data for reconstruction. A theoretical overview of frame-based representation of scalar time-harmonic fields is presented in [19], with an extension to electromagnetic fields in [23], and for time-dependent scalar fields in [20].…”